What is the derivative of : y=sin(ln(sqrt(6x^2*lnx^2)))

Since the given function is the result of composition of
more than 2 functions, we'll apply chain rule:

y' =
[sin(ln(sqrt(6x^2*lnx^2)))]'*(ln(sqrt(6x^2*lnx^2)))'*(sqrt(6x^2*lnx^2))'*(6x^2*lnx^2)'

The
last factor has to be differentiated applying the product rule and chain
rule.

y' =
[cos(ln(sqrt(6x^2*lnx^2)))]*[1/(sqrt(6x^2*lnx^2))]*[1/2*sqrt(6x^2*lnx^2)]*[(6x^2)'*lnx^2
+ (6x^2)*(lnx^2)']

y' = [cos(ln(sqrt(6x^2*lnx^2))]*(12x*ln
x^2 + 6x^2*2x/x^2)/2*(6x^2*lnx^2)

y' =
12x*(1+lnx^2)*[cos(ln(sqrt(6x^2*lnx^2))]/12x^2*lnx^2

y'
= (1+lnx^2)*[cos(ln(sqrt(6x^2*lnx^2))]/x*lnx^2